Find the Domain and Range f(theta)=1+sec(pitheta)

$f (θ) = 1 + \sec (π θ)$

Step 1

Set the argument in $\sec (π θ)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$π θ = \frac{π}{2} + π n$ , for any integer $n$

Step 2

Divide each term in $π θ = \frac{π}{2} + π n$ by $π$ and simplify.

Divide each term in $π θ = \frac{π}{2} + π n$ by $π$ .

$\frac{π θ}{π} = \frac{\frac{π}{2}}{π} + \frac{π n}{π}$

Simplify the left side.

Cancel the common factor of $π$ .

Cancel the common factor.

$\frac{π θ}{π} = \frac{\frac{π}{2}}{π} + \frac{π n}{π}$

Divide $θ$ by $1$ .

$θ = \frac{\frac{π}{2}}{π} + \frac{π n}{π}$

Simplify the right side.

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{π}{2} \cdot \frac{1}{π} + \frac{π n}{π}$

Cancel the common factor of $π$ .

Cancel the common factor.

$θ = \frac{π}{2} \cdot \frac{1}{π} + \frac{π n}{π}$

Rewrite the expression.

$θ = \frac{1}{2} + \frac{π n}{π}$

Cancel the common factor of $π$ .

Cancel the common factor.

$θ = \frac{1}{2} + \frac{π n}{π}$

Divide $n$ by $1$ .

$θ = \frac{1}{2} + n$

Step 3

The domain is all values of $θ$ that make the expression defined.

Set-Builder Notation:

${θ | θ \neq \frac{1}{2} + n}$ , for any integer $n$

Step 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, 0] \cup [2, \infty)$

Set-Builder Notation:

${y | y \leq 0, y \geq 2}$

Step 5

Determine the domain and range.

Domain: ${θ | θ \neq \frac{1}{2} + n}$ , for any integer $n$

Range: $(- \infty, 0] \cup [2, \infty), {y | y \leq 0, y \geq 2}$

Step 6